One more difference that is two differences after the

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one more difference, that is two differences after the seasonal difference, we see in the following it makes the variance higher which indicates overdifferencing: d e a t h s d i f f 1 2 d i f f 2 < - diff ( d e a t h s d i f f 1 2 d i f f 1 , lag =1) var ( d e a t h s d i f f 1 2 d i f f 2 ) The variance is given by var ( d e a t h s d i f f 1 2 d i f f 2 ) ## 428291.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Choose a dataset that you will be interested to analyze for your class final project. URLs of time series libraries are posted on Gaucho Space. Provide the following information about the project: (a) Data set description: briefly describe the data set you plan to use in your project. (b) Motivation and objectives: briefly explain why this data set is interesting or important. Provide a clear description of the problem you plan to address using this dataset (for exam- ple to forecast). (c) Plot and examine the main features of the graph, checking in particular whether there is (i) a trend; (ii) a seasonal component, (iii) any apparent sharp changes in behavior. Explain in detail. (d) Use any necessary transformations to get stationary series. Give a detailed explanation to justify your choice of a particular procedure. If you have used transformation, justify why. If you have used differencing, what lag did you use? Why? Is your series stationary now? (e) Plot and analyze the ACF and PACF to preliminary identify your model(s): Plot ACF/- PACF. What model(s) do they suggest? Explain your choice of p and q here. Note that you may change the project dataset in the future. This question is designed to help you start planning your the project. Please include plots of corresponding theoretical acfs and the corresponding R code. This is an individual assignment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The following problem is for students enrolled in PSTAT 274 ONLY 6. Find the ACVF and ACF functions for { X t } when X t = Φ X t - 4 + Z t , | Φ | < 1. Solution. First, write the autocovariance function for X t : γ X ( k ) = E [ X t X t - k ] = E [(Φ X t - 4 + Z t ) X t - k ] = ( Φ γ X ( k - 4) , k 1 Φ γ X ( k - 4) + E [ Z t X t ] k = 0 For k 1 we have: γ X ( k ) = Φ γ X ( k - 4) . Take k = 4 h, h = 1 , 2 , . . . to see that γ X (4) = Φ γ X (0) , γ X (8) = Φ γ X (4) = Φ 2 γ X (0) , γ X (4 h ) = Φ h γ X (0) , h = 1 , 2 , 3 , . . . . To find out the ACVF when k 6 = 4 , and to find E [ Z t X t ] we need to re-express the model as an infinite order moving average (similar technique was used in 4.1 of Lecture 4 in derivation of ACF of AR(1).) X t = (1 - Φ B 4 ) - 1 Z t = X j =0 B 4 ) j Z t = X j =0 Φ j Z t - 4 j . This study resource was shared via CourseHero.com
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It then follows that γ X ( k ) = E [ X t X t - k ] = E h j =0 Φ j Z t - 4 j (∑ i =0 Φ i Z t - k - 4 i ) i = j =0 i =0 Φ i + j E [ Z t - 4 j Z t - k - 4 i ] If k is not a multiple of 4, then t - 4 j 6 = t - k - 4 i for all i and j . Thus, we have that E [ Z t - 4 j Z t - k - 4 i ] = 0 and γ X ( k ) = 0 , consequently.
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